# Definition: Conditional differential entropy

**Index:**The Book of Statistical Proofs ▷ General Theorems ▷ Information theory ▷ Differential entropy ▷ Conditional differential entropy

**Definition:** Let $X$ and $Y$ be continuous random variables with possible outcomes $\mathcal{X}$ and $\mathcal{Y}$ and probability density functions $p(x)$ and $p(y)$. Then, the conditional differential entropy of $Y$ given $X$ or, differential entropy of $Y$ conditioned on $X$, is defined as

where $\mathrm{h}(Y \vert X=x)$ is the (marginal) differential entropy of $Y$, evaluated at $x$.

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**Metadata:**ID: D34 | shortcut: dent-cond | author: JoramSoch | date: 2020-03-21, 12:27.